I'm married to a Finnish woman and live in Finland. My wife became Muslim five years ago. She pronounces the Shahadatain a lot, every time alone or if there is other people and she tries hard to learn the Qur'an in Finnish.My question is who does she need to officially become Muslim. Does she need any official document to make Umrah or Hajj, even in case of death, because she wants to be buried with other Muslims, for example, in Tunisia or some other Muslim country? The problem is the Tunisian government wants such a document.

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Whenever a person utters the Shahadatain (bearing two witnesses) sincerely and faithfully and is determined to implementing the basics of Sharia such as the prayers, he becomes a Muslim. He no longer needs to register himself as a Muslim or take any certificate for that.
However, if he cannot perform Hajj or Umrah or could be buried in a graveyard of non-Muslim due to what is written in official papers such as his passport, etc., then he should change his religion in those papers to the religion of Islam. These mentioned rulings are common for men and women.
We ask Allah to make His believers stead-fast on the right path.
Allah knows best.